Global Clifford Closure

Global Clifford Closure for the PBG Spinor Sector

29 Jun 2025 · v0.1

Purpose – Show that the γ–matrices built from the internal S³ frame in the spinor-extended PBG satisfy
${γ^{μ} (x), γ^{ν} (x)} = 2 η^{μ ν} 1$
everywhere in space-time. No Pauli or Dirac matrices are “inserted by hand”; they appear once, at a single reference point, as a choice of representation. Thereafter the Clifford property follows from the orthonormal frame itself.

1 · Internal orthonormal frame

For each spacetime point $x^{μ}$ define

e_{\hat{a}}^{b} (x), \hat{a} = 0, 1, 2, 3, e_{\hat{a}}^{b} e_{\hat{b}}^{c} δ_{b c} = η_{\hat{a} \hat{b}}, N^{b} (x) e_{\hat{a} b} (x) = 0.

The hedgehog background chosen in § 1 of the main note provides an explicit construction of $e_{\hat{a}}^{b}$ .

2 · Definition of the γ–matrices

Choose one constant representation $Γ^{\hat{a}}$ of the abstract Clifford algebra:

Γ^{\hat{a}} Γ^{\hat{b}} + Γ^{\hat{b}} Γ^{\hat{a}} = 2 η^{\hat{a} \hat{b}} .

Then set, at every point,

γ^{μ} (x) \equiv e_{\hat{a}}^{μ} (x) Γ^{\hat{a}} .

3 · Global Clifford proof

\begin{aligned} γ^{μ} (x) γ^{ν} (x) + γ^{ν} (x) γ^{μ} (x) & = e_{\hat{a}}^{μ} e_{\hat{b}}^{ν} (Γ^{\hat{a}} Γ^{\hat{b}} + Γ^{\hat{b}} Γ^{\hat{a}}) \\ = 2 e_{\hat{a}}^{μ} e^{\hat{a} ν} 1 = 2 g^{μ ν} (x) 1 . \end{aligned}

Since the coherence background is Minkowskian until macroscopic
envelopes curve it, $g^{μ ν} (x) = η^{μ ν}$ .
Hence

{γ^{μ} (x), γ^{ν} (x)} = 2 η^{μ ν} 1 \forall x .

No hidden insertions: the orthonormality of $e_{\hat{a}}^{μ}$
propagates the algebra globally.

4 · Spin connection preview

The emergent spin connection is

ω_{μ}^{\hat{a} \hat{b}} = e_{c}^{\hat{a}} \partial_{μ} e^{\hat{b} c}, Γ_{μ} = \frac{1}{4} ω_{μ}^{\hat{a} \hat{b}} Γ_{\hat{a}} Γ_{\hat{b}} .

In the next appendix we will show that
$D_{μ} = \partial_{μ} + Γ_{μ}$
gives the covariant derivative that appears in the full Dirac equation for the hedgehog’s transverse mode $χ$ .

5 · Symbolic-algebra notebook

A SymPy / Cadabra file that constructs $e_{\hat{a}}^{μ} (x)$ , builds $γ^{μ} (x)$ , and verifies the Clifford relation can be downloaded here:
[spinor_frame_clifford_proof.ipynb](./attachments/spinor_frame_clifford_proof.ipynb).

(Runs in ≈ 3 s and prints “All anticommutators passed.”)

End of Appendix AC